Paper: arXiv 2603.12767

Authors: Mihaela-Adriana Nistor, Ionel Popescu

Abstract

In this paper we introduce a generalization of classical risk measures in which the risk is represented by a step function taking two values, corresponding to two endogenously determined market regimes. This extends the traditional framework where risk measures map random variables to single real numbers. For the quadratic loss function, we study the optimization problem of determining the optimal regime threshold and corresponding values. In the case of log-concave distributions we give conditions for the uniqueness of the regime changing. We treat the case of one dimension and also of multi-dimensions for elliptic distributions. We demonstrate the necessity of convexity through counterexamples.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a strong theoretical framework with rigorous proofs on log-concave and elliptic distributions for multi-regime risk measures. However, it is purely mathematical and lacks empirical backtesting, datasets, or code for real-world validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Generalize risk measures using two-step regime-switching approximations] --> B[Inputs: Quadratic loss function & Log-concave / Elliptic distributions]
    B --> C[Methodology: Formulate optimization problem for threshold & regime values]
    C --> D[Analysis: Evaluate 1D log-concave & multi-D elliptic cases]
    D --> E[Counterexamples: Test necessity of convexity conditions]
    E --> F[Outcomes: Uniqueness conditions for regime changes & proof that convexity is required]

Cited References (55)

  1. Quantitative risk management: concepts, techniques and tools-revised edition (2015) — link
  2. What is the best risk measure in practice? A comparison of standard measures (2013) — link
  3. Coherent measures of risk (1999) — link
  4. Coherence and elicitability (2016) — link
  5. Expected shortfall: a natural coherent alternative to value at risk (2002) — link
  6. A new approach to the economic analysis of nonstationary time series and the business cycle (1989) — link
  7. Regime switches in interest rates (2002) — link
  8. Autoregressive conditional heteroskedasticity and changes in regime (1994) — link
and 47 more
  1. State-Space Models with Regime Switching: Classical and Gibbs-Sampling Approaches with Applications (1999) — link
  2. Dynamic risk measures (2010) — link
  3. Log-Concavity and Strong Log-Concavity: A Review (2014) — link
  4. Conditional value-at-risk and average value-at-risk: Estimation and asymptotics (2012) — link
  5. Optimization of conditional value-at-risk (2000) — link
  6. Convex measures of risk and trading constraints (2002) — link
  7. Stochastic Finance: An Introduction in Discrete Time (2016) — link
  8. Lectures on Stochastic Programming: Modeling and Theory (2014) — link
  9. Application of coherent risk measures to capital requirements in insurance (1999) — link
  10. On elicitable risk measures (2015) — link
  11. Measures of systemic risk (2017) — link
  12. Risk management with expectiles (2017) — link
  13. The axiomatic approach to risk measures for capital determination (2015) — link
  14. Interpreting expectiles (2022) — link
  15. Dynamic coherent risk measures (2004) — link
  16. Convex and coherent risk measures (2010) — link
  17. Coherent risk measures in inventory problems (2007) — link
  18. Pricing with coherent risk (2008) — link
  19. Conditional and dynamic convex risk measures (2005) — link
  20. Coherent risk measures on general probability spaces (2002) — link
  21. Can a coherent risk measure be too subadditive? (2008) — link
  22. Entropic value-at-risk: A new coherent risk measure (2012) — link
  23. Putting order in risk measures (2002) — link
  24. Spectral measures of risk: A coherent representation of subjective risk aversion (2002) — link
  25. Distributionally robust optimization under moment uncertainty with application to data-driven problems (2010) — link
  26. Wasserstein distributionally robust optimization: Theory and applications in machine learning (2019) — link
  27. Log-concave probability and its applications (2005) — link
  28. Strong log-concavity is preserved by convolution (2010) — link
  29. Coherent multiperiod risk measurement (2002) — link
  30. Measures of risk (2002) — link
  31. Coherent multiperiod risk adjusted values and Bellman’s principle (2007) — link
  32. Shortfall as a risk measure: properties, optimization and applications (2004) — link
  33. Robustness and sensitivity analysis of risk measurement procedures (2010) — link
  34. Conditional value-at-risk in portfolio optimization: Coherent but fragile (2011) — link
  35. Convex measures of risk and trading constraints (2002) — link
  36. Extreme value theory for risk managers (1999) — link
  37. Conditional value-at-risk for general loss distributions (2002) — link
  38. Worst-case value at risk of nonlinear portfolios (2013) — link
  39. Expected shortfall as a tool for financial risk management (2001) — link
  40. Measuring market risk (2007) — link
  41. Unique optimal partitions of distributions and connections to hazard rates and stochastic ordering (2006) — link
  42. Sufficient conditions for uniqueness of a locally optimal quantizer for a class of convex error weighting functions (1982) — link
  43. Monotony of L loyd’s method II for log-concave density and convex error weighting function (1984) — link
  44. Optimal grouping, spacing, stratification, and piecewise constant approximation (1988) — link
  45. Optimal Partitioning of Probability Distributions under General Convex Loss Functions in Selective Assembly (2011) — link
  46. Signaling Games for Log-Concave Distributions: Number of Bins and Properties of Equilibria (2022) — link
  47. Coarse Information Design (2025) — link

0 of 55 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.